The simultaneous parametric plot of S(x) and C(x) is the Euler spiral (also known as the Cornu spiral or clothoid). Recently, they have been used in the design of highways and other engineering projects.
Some authors, including Abramowitz and Stegun, (eqs 7.3.1 – 7.3.2) use for the argument of the integrals defining S(x) and C(x), which changes their limits to 0.5 and the arc length for the first spiral turn to 2 (at ), all smaller by a factor . The alternative functions are also called Normalized Fresnel integrals.
The Euler spiral, also known as Cornu spiral or clothoid, is the curve generated by a parametric plot of against . The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering.
From the definitions of Fresnel integrals, the infinitesimals and are thus:
Thus the length of the spiral measured from the origin can be expressed as
That is, the parameter is the curve length measured from the origin , and the Euler spiral has infinite length. The vector also expresses the unittangent vector along the spiral, giving . Since t is the curve length, the curvature can be expressed as
And the rate of change of curvature with respect to the curve length is
An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering: If a vehicle follows the spiral at unit speed, the parameter in the above derivatives also represents the time. That is, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.
Sections from Euler spirals are commonly incorporated into the shape of roller-coaster loops to make what are known as clothoid loops.
around the boundary of the sector-shaped region in the complex plane formed by the positive x-axis, the bisector of the first quadrant y = x with x ≥ 0, and a circular arc of radius R centered at the origin.
As R goes to infinity, the integral along the circular arc tends to 0, the integral along the real axis tends to the half Gaussian integral
Note too that because the integrand is an entire function on the complex plane, its integral along the whole contour is zero. Overall, we must have
where denotes the bisector of the first quadrant, as in the diagram. To evaluate the right hand side, parametrize the bisector as where r ranges from 0 to . Note that the square of this expression is just . Therefore, substitution gives the right hand side as
where we have written to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting and then equating real and imaginary parts produces the following system of two equations in the two unknowns :
which reduces to Fresnel integrals if real or imaginary parts are taken:
The leading term in the asymptotic expansion is
For m=0, the imaginary part of this equation in particular is
with the left-hand side converging for a>1 and the right-hand side being its analytical extension to the whole plane less where lie the poles of .
The Kummer transformation of the confluent hypergeometric function is
For computation to arbitrary precision, the power series is suitable for small argument. For large argument, asymptotic expansions  converge faster. Continued fraction methods may also be used. 
For computation to particular target precision, other approximations have been developed. Cody  developed a set of efficient approximations based on rational functions that give relative errors down to 6981199999999999999♠2×10−19. A FORTRAN implementation of the Cody approximation that includes the values of the coefficients needed for implementation in other languages was published by van Snyder. Boersma developed an approximation with error less than 6991160000000000000♠1.6×10−9. 
The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects. More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see track transition curve. Other applications are roller coasters or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.